Credit Risk Modelling : A Primer
By: A V Vedpuriswar
June 14, 2014
Introduction to Credit Risk Modelling
Credit risk modeling helps to estimate how much credit is 'at
risk' due to a default or changes in credit risk factors.
By doing so, it enables managers to price the credit risks they
face more effectively.
It also helps them to calculate how much capital they need to
set aside to protect against such risks.
1
Market Risk vs Credit Risk Modelling
Compared to market risk modeling, credit risk modeling is
relatively new.
Credit risk is more contextual.
The time horizon is usually longer for credit risk.
Legal issues are more important in case of credit risk.
The upside is limited while the downside is huge.
If counterparty defaults, while the contract has negative value,
the solvent party typically cannot walk away from the contract.
But if the defaulting party goes bankrupt, while contract has a
positive value, only a fraction of the funds owed will be received.
2
Data
There are serious data limitations.
Market risk data are plentiful.
But default/bankruptcy data are rare.
3
Liquidity
Market prices are readily available for instruments that give rise
to market risk.
However, most credit instruments don't have easily observed
market prices.
There is less liquidity in the price quotes for bank loans,
compared to interest rate instruments or equities.
This lack of liquidity makes it very difficult to price credit risk for a
particular obligor in a mark-to-market approach.
To overcome this lack of liquidity, credit risk models must
sometimes use alternative types of data (historical loss data).
4
Distribution of losses
Market risk is often modeled by assuming that returns follow a
normal distribution though sometimes it does not hold good.
The normal distribution, however, is completely inappropriate for
estimating credit risk.
 Returns in the global credit markets are heavily skewed to the
downside and are therefore distinctly non-normal.
Banks' exposures are asymmetric in nature.
There is limited upside but large downside.
The distribution exhibits a fat tail.
5
Correlation & Diversification
Diversification is the main tool for reducing credit risk.
For most obligors, hedges are not available in the market.
But there are limits to diversification.
A loan portfolio might look well diversified by its large number
of obligors.
But there might still be concentration risk caused by a large
single industry/country exposure.
Also correlations can dramatically shoot up in a crisis.
6
Expected, unexpected and stress losses
7
Expected Loss
The expected loss (EL) is the amount that an institution
expects to lose on a credit exposure over a given time horizon.
EL = PD x LGD x EAD
If we ignore correlation between the LGD variable, the EAD
variable and the default event, the expected loss for a portfolio
is the sum of the individual expected losses.
How should we deal with expected losses?
In the normal course of business, a financial institution can set
aside an amount equal to the expected loss as a provision.
Expected loss can be built into the pricing of loan products.
8
Unexpected loss
Unexpected loss is the amount by which potential credit losses
might exceed the expected loss.
Traditionally, unexpected loss is the standard deviation of the
portfolio credit losses.
But this is not a good risk measure for fat-tail distributions,
which are typical for credit risk.
To minimize the effect of unexpected losses, institutions are
required to set aside a minimum amount of regulatory capital.
Apart from holding regulatory capital, however, many
sophisticated banks also estimate the necessary economic
capital to sustain these unexpected losses.
9
Stress Losses
Stress losses are those that occur in the tail region of the
portfolio loss distribution.
They occur as a result of exceptional or low probability events
(a 0.1% or 1 in 1,000 probability in the distribution below).
While these events may be exceptional, they are also
plausible and their impact is severe.
10
Measuring Credit loss
In simple terms, a credit loss can be described as a decrease
in the value of a portfolio over a specified period of time.
So we must estimate both current value and the future value
of the portfolio at the end of a given time horizon.
There are two conceptual approaches for measuring credit
loss:
– default mode paradigm
– mark-to-market paradigm
11
Default mode paradigm
A credit loss occurs only in the event of default..
This approach is sometimes referred to as the two-state model.
The borrower either does or does not default.
If no default occurs, the credit loss is obviously zero.
If default occurs, exposure at default and loss given default must
be estimated.
12
Mark-to-market (MTM) paradigm
Here , a credit loss occurs if:
– the borrower defaults
– the borrower's credit quality deteriorates (credit migration)
This is therefore a multi-state paradigm.
There can be an economic impact even if there is no default.
A true mark-to-market approach would take market-implied
values in different non-defaulting states.
 However, because of data and liquidity issues, some banks use
internal prices based on loss experiences.
13
Classification of other approaches
Top down vs Bottom Up
Structural vs Reduced form
Conditional vs Unconditional
14
Mark-to-market paradigm approaches
There are two well-known approaches in the mark-to-market
paradigm :
– the discounted contractual cash flow approach
– the risk-neutral valuation approach
15
Discounted Contractual Cashflow Approach
The current value of a non-defaulted loan is measured as the
present value of its future cash flows.
The cash flows are discounted using credit spreads which are
equal to market-determined spreads for obligations of the same
grade.
If external market rates cannot be applied, spreads implied by
internal default history can be used.
The future value of a non-defaulted loan is dependent on the
risk rating at the end of the time horizon and the credit spreads
for that rating.
Therefore, changes in the value of the loan are the result of credit
migration or changes in market credit spreads.
In the event of a default, the future value is determined by the
recovery rate, as in the default mode paradigm.
16
Risk-Neutral Valuation Approach
 This approach is derived from derivatives pricing theory.
 Prices are an expectation of the discounted future cash flows in a risk-neutral
market.
 These default probabilities are therefore called risk-neutral default probabilities
and are derived from the asset values in a risk-neutral option pricing approach.
 Each cash flow in the risk-neutral approach depends on there being no default.
 For example, if a payment is contractually due on a certain date, the lender
receives the payment only if the borrower has not defaulted by this date.
 If the borrower defaults before this date, the lender receives nothing.
 If the borrower defaults on this date, the value of the payment to the lender is
determined by the recovery rate (1 - LGD rate).
 The value of a loan is equal to the sum of the present values of these cash
flows.
17
Structural and Reduced Form Models
18
Structural Models

Probability of default is determined by
– the difference between the current value of the firm's assets
and liabilities, and
– by the volatility of the assets.
Structural models are based on variables that can be observed
over time in the market.
Asset values are inferred from equity prices.
Structural models are difficult to use if the capital structure is
complicated and asset prices are not easily observable.
19
Reduced Form Models
Reduced form models do not attempt to explain default events.
Instead, they concentrate directly on default probability.
Default events are assumed to occur unexpectedly due to one or
more exogenous events (observable and unobservable),
independent of the borrower's asset value.
Observable risk factors include changes in macroeconomic
factors such as GDP, interest rates, exchange rates, inflation.
Unobservable risk factors can be specific to a firm, industry or
country.
Correlations among PDs for different borrowers are considered to
arise from the dependence of different borrowers on the behavior
of the underlying background factors.
20
Reduced Form Models
 Default in the reduced form approach is assumed to follow a Poisson
distribution.
 A Poisson distribution describes the number of events of some phenomenon
(in this case, defaults) taking place during a specific period of time.
 It is characterized by a rate parameter (t), which is the expected number of
arrivals that occur per unit of time.
 In a Poisson process, arrivals occur one at a time rather than simultaneously.
 And any event occurring after time t is independent of an event occurring
before time t.
 It is therefore relevant for credit risk modeling –
– There is a large number of obligors.
– The probability of default by any one obligor is relatively small.
– It is assumed that the number of defaults in one period is independent of
the number of defaults in the following period.
21
Correlations
The modeling of the covariation between default probability (PD)
and exposure at default (EAD) is particularly important in the
context of derivative instruments, where credit exposures are
particularly market-driven.
A worsening of exposure may occur due to market events that tend
to increase EAD while simultaneously reducing a borrower's ability
to repay debt (that is, increasing a borrower's probability of default).
22
Correlations
There may also be correlation between exposure at default (EAD)
and loss given default (LGD).
 For example, LGDs for borrowers within the same industry may
tend to increase during periods when conditions in that industry
are deteriorating (or vice-versa).
The ability of banks to model these correlations, however, has
been restricted due to data limitations and technical issues.
LGD is frequently modeled as a fixed percentage of EAD, with
actual percentage depending on the seniority of the claim.
In practice, LGD is not constant.
So attempts have been made to model it as a random variable or
to treat it as being dependent on other variables.
23
Credit Risk Models
Merton
Moody's KMV
Credit Metrics
Credit Risk+
Credit Portfolio View
25
Merton and KMV models
26
The Merton Model
This model assumes that the firm has made one single issue
of zero coupon debt and equity.
Let V be value of the firm’s assets, D value of debt.
When debt matures, debt holders will receive the full value of
their debt, D provided V > D.
Equity holders will receive V-D.
If V < D, debt holders will receive only a part of the sums due
and equity holders will receive nothing.
Value received by debt holders at time T = D – max {D-VT, 0}
27
The Payoff from Debt
Examine : D – max {D-VT, 0}
D is the pay off from investing in a default risk free instrument.
On the other hand, - max {D-VT, 0} is the pay off from a short
position in a put option on the firm’s assets with a strike price
of D and a maturity date of T
Thus risky debt ☰ long default risk free bond + short put
option with strike price D
28
Value of the put
Value of the put completely determines the price differential
between risky and riskless debt.
A higher value of the put increases the price difference
between risky and riskless bonds.
As volatility of firm value increases, the spread on the risky
debt increases and the value of the put increases.
29
Value of equity
Let E be the value of the firm’s equity.
Let E be the volatility of the firm’s equity.
Claim of equity
= VT – D if VT ≥ D
= 0 otherwise
The pay off is the same as that of a long call with strike price D.
30
Valuing the put option
Assume the firm value follows a lognormal distribution with
constant volatility, .
Let the risk free rate, r be also constant.
Assume dV = µV dt + V dz ( Geometric Brownian motion)
The value of the put, p at time, t is given by:
p = K e-r(T-t) N (-d2) – S N(-d1)
p = D e-r(T-t) N (-d1 + T-t) – V t N(-d1)
d1 = [1/  T-t] [ln (V t /D) + (r+ ½ 2 (T-t)]
31
Valuing the call option
The value of the call is a function of the firm value and firm
volatility.
Firm volatility can be estimated from equity volatility.
The value of the call can be calculated by:
c = S N(d1) – K e-r(T-t) N (d2 )
c = Vt N(d1) – D e-r(T-t) N (d1 - T-t)
32
Problem
 The current value of the firm is $60 million and the value of the zero coupon
bond to be redeemed in 3 years is $50 million. The annual interest rate is
5% while the volatility of the firm value is 10%. Using the Merton Model,
calculate the value of the firm’s equity.
 Value of equity = Ct = Vt x N(d) – De-r(T-t) x N (d-T-t)

d = [1/  T-t] [ln (V t /D) + (r+ ½ 2) (T-t)]

Ct
=
60 x N (d) – (50)e-(.05)(3) x N [d-(.1)3]

d
=
[.1823 +( .05+.01/2)(3)]/.17321
=
.3473/ .17321 = 2.005

Ct
=
60 N (2.005) – (50) (.8607) N (2.005 - .17321)

=
60 N (2.005) – (43.035) N (1.8318)

=
(60) (.9775) – (43.035) (.9665)

=
$17.057 million
V = value of firm,

D = face value of zero coupon debt
 = firm value volatility, r =
interest rate
33
Problem

 Dt
In the earlier problem, calculate the value of the firm’s
=
debt.
De-r(T-t) – pt

=
50e-.05(3) – pt

=
43.035 – pt
 Based on put call parity

pt
=
Ct + De-r(T-t) – V
 Or
pt
=
17.057 + 43.035 – 60
= .092

Dt
=
43.035 - .092
= $42.943 million
 Alternatively, value of debt

=
Firm value – Equity value

=
$42.943 million
= 60 – 17.057
34
Problem
The value of an emerging market firm’s asset is $20 million.
The firm’s sole liability consists of a pure discount bond with
face value of $15 million and one year remaining until
maturity.
At the end of the next year, the value of firm’s assets will
either be $40 million or $10 million.
The riskless interest rate is 20 percent.
Compute the value of the firm’s equity and the value of the
firm’s debt.
35
Solution
Define V as the value of the firm’s assets. In a binomial
framework,
V, T, u = 40
 V, T - 1 = 20
V, T, d = 10
Define E as the value of the firm’s equity, and K as the
face value of the firm’s debt. K = 15. then
ET, u = 25 = 40 - 15

ET - 1
ET, d = 0
36
Cont…
Let current asset value be V.
At the end of a period, asset values can be V (1+ u) ie 40 or
V(1+d) ie 10.
If the firm’s assets have an uptick, then u = [40-20)/20] = 1.0.
The value of d is d = [(20-40)/40] = - 0.5.
Therefore, with r = 0.20,
p
r  d 0.2  0.5

 0.466667
ud
1  0.5
 E  pET ,u  (1  p) ET ,d  (0.46667)(25)  (0.5333)(0) = 9.72
T 1
1 r
1.2
The value of the firm’s assets is currently 20, V = E + D,
Value of firm’s debt = 20-9.72 = 10.28
37
Complex capital structures
In real life, capital structures may be more complex.
There may be multiple debt issues differing in
– maturity,
– size of coupons
– seniority.
Equity then becomes a compound option on firm value.
Each promised debt payment gives the equity holders the right
to proceed to the next payment.
If the payment is not made, the firm is in default.
After last but one payment is made, Merton model applies.
38
KMV Model
 Default tends to occur when the market value of the firm’s
assets drops below a critical point that typically lies
– Below the book value of all liabilities
– But above the book value of short term liabilities
 The model identifies the default point d used in the
computations.
 The KMV model assumes that there are only two debt issues.
 The first matures before the chosen horizon and the other
matures after that horizon.
The probability of exercise of the put option is the probability of
default.
 The distance to default is calculated as:
lnV  l nD  (r   2 / 2)T
 T
39
KMV Model
The distance to default, d2 is a proxy measure for the
probability of default.
As the distance to default decreases, the company becomes
more likely to default.
As the distance to default increases, the company becomes
less likely to default.
The KMV model, unlike the Merton Model does not use a
normal distribution.
Instead, it assumes a proprietary algorithm based on historical
default rates.
40
KMV Model
Using the KMV model involves the following steps:
– Identification of the default point, D.
– Identification of the firm value V and volatility 
– Identification of the number of standard deviation moves
that would result in firm value falling below D.
– Use KMV database to identify proportion of firms with
distance-to-default, δ who actually defaulted in a year.
– This is the expected default frequency.
– KMV takes D as the sum of the face value of the all short
term liabilities (maturity < 1 year) and 50% of the face value
of longer term liabilities.
41
Problem
 Consider the following figures for a company. What is the probability of
default?
– Book value of all liabilities
: $2.4 billion
– Estimated default point, D
: $1.9 billion
– Market value of equity
: $11.3 billion
– Market value of firm
: $13.8 billion
– Volatility of firm value
: 20%
Solution
 Distance to default (in terms of value)
= 13.8 – 1.9 = $11.9 billion
 Standard deviation
= (.20) (13.8) = $2.76 billion
 Distance to default (in terms of standard deviation) = 11.9/2.76 = 4.31
 We now refer to the default database. If 5 out of 100 firms with distance to
default = 4.31 actually defaulted, probability of default = .05
42
Problem
Given the following figures, compute the distance to default:
– Book value of liabilities
:
$5.95 billion
– Estimated default point
:
$4.15 billion
– Market value of equity
:
$ 12.4 billion
– Market value of firm
:
$18.4 billion
– Volatility of firm value
:
24%
Solution
Distance to default (in terms of value) = 18.4 – 4.15 = $14.25
billion
Standard deviation
=
Distance to default (in terms of )
(.24) (18.4) = $4.416 billion
= 14.25/4.42 = 3.23
43
Portfolio Credit Risk Models : Conclusion
Top-down models group credit risk single statics.
They aggregate many sources of risk viewed as
homogeneous into an overall portfolio risk, without going into
the details of individual transactions.
This approach is appropriate for retail portfolios with large
numbers of credits, but less so for corporate or sovereign
loans.
Even within retail portfolios, top-down models may hide
specific risks, by industry or geographic location.
44
Bottom up models
Bottom-up models account for features of each instrument.
This approach is most similar to the structural decomposition
of positions that characterizes market VAR systems.
It is appropriate for corporate and capital market portfolios.
Bottom-up models are most useful for taking corrective action,
because the risk structure can be reverse-engineered to
modify this risk profile.
45
Default mode and mark to market
Default-mode models consider only outright default as a
credit event.
Hence any movement in the market value of the bond or
in the credit rating is irrelevant.
Mark-to-market models consider changes in market
values and ratings changes, including defaults.
They provide a better assessment of risk, which is
consistent with the holding period defined in terms of the
liquidation period.
46
Conditional, structural, reduced form models
Conditional models incorporate changing macroeconomic factors
into the default probability through a functional relationship.
The rate of default increases in a recession.
Structural models explain correlations by the joint movements of
assets – for example, stock prices.
For each obligator, this price is the random variable that
represents movements in default probabilities.
Reduced-form models explain correlations by assuming a
particular functional relationship between the default probability
and “background factor.”
For example, the correlation between defaults across obligors can
be modeled by the loadings on common risk factors – say,
industrial and country.
47
Comparison of Credit Risk Models
CreditMetrics
CreditRisk+
KMV
CreditPf.View
Originator
J P Morgan
Credit Suisse
KMV
McKinsey
Model type
Bottom-up
Bottom-up
Bottom-up
Top-down
Risk definition
Market value
(MTM)
Default losses
(DM)
Default losses
(MTM/DM)
Market value
(MTM)
Risk drivers
Asset values
Default rates
Asset values
Macro factors
Credit events
Rating
change/default
Default
Continuous
default prob.
Rating
change/default
Probability
Unconditional
Unconditional
Conditional
Conditional
Volatility
Constant
Variable
Variable
Variable
Correlation
From equities
(structural)
Default process
(reduced-form)
From equities
(structural)
From macro
factors
Recovery rates Random
Constant within
band
Random
Random
Solution
Analytic
Analytic
Simulation
Simulation/
analytic
48